Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Get immediate and reliable answers to your questions from a community of experienced experts on our platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
Sure! Let's tackle each part step-by-step.
### 1. Remove the greatest common factor (GCF) and express each expression as the product of a GCF and a polynomial.
#### a. [tex]\(7 b^2 + 7 b\)[/tex]
1. Identify the GCF: The GCF of [tex]\(7 b^2\)[/tex] and [tex]\(7 b\)[/tex] is [tex]\(7 b\)[/tex].
2. Factor out the GCF: [tex]\(7 b(b + 1)\)[/tex]
So, the expression is [tex]\(7 b(b + 1)\)[/tex].
#### b. [tex]\(24 a b - 18 a b^2\)[/tex]
1. Identify the GCF: The GCF of [tex]\(24 a b\)[/tex] and [tex]\(18 a b^2\)[/tex] is [tex]\(6 a b\)[/tex].
2. Factor out the GCF: [tex]\(6 a b(4 - 3 b)\)[/tex]
So, the expression is [tex]\(6 a b(4 - 3 b)\)[/tex].
#### c. [tex]\(5 a^2 b^2 - 35 a b\)[/tex]
1. Identify the GCF: The GCF of [tex]\(5 a^2 b^2\)[/tex] and [tex]\(35 a b\)[/tex] is [tex]\(5 a b\)[/tex].
2. Factor out the GCF: [tex]\(5 a b(a b - 7)\)[/tex]
So, the expression is [tex]\(5 a b(a b - 7)\)[/tex].
#### d. [tex]\(8 x^3 - 10 x^2 - 14 x\)[/tex]
1. Identify the GCF: The GCF of [tex]\(8 x^3\)[/tex], [tex]\(10 x^2\)[/tex], and [tex]\(14 x\)[/tex] is [tex]\(2 x\)[/tex].
2. Factor out the GCF: [tex]\(2 x(4 x^2 - 5 x - 7)\)[/tex]
So, the expression is [tex]\(2 x(4 x^2 - 5 x - 7)\)[/tex].
#### e. [tex]\(36 x^6 + 45 x^4 - 18 x^2\)[/tex]
1. Identify the GCF: The GCF of [tex]\(36 x^6\)[/tex], [tex]\(45 x^4\)[/tex], and [tex]\(18 x^2\)[/tex] is [tex]\(9 x^2\)[/tex].
2. Factor out the GCF: [tex]\(9 x^2(4 x^4 + 5 x^2 - 2)\)[/tex]
So, the expression is [tex]\(9 x^2(4 x^4 + 5 x^2 - 2)\)[/tex].
#### f. [tex]\(40 a^2 - 16 a b - 24 a\)[/tex]
1. Identify the GCF: The GCF of [tex]\(40 a^2\)[/tex], [tex]\(16 a b\)[/tex], and [tex]\(24 a\)[/tex] is [tex]\(8 a\)[/tex].
2. Factor out the GCF: [tex]\(8 a(5 a - 2 b - 3)\)[/tex]
So, the expression is [tex]\(8 a(5 a - 2 b - 3)\)[/tex].
#### g. [tex]\(8 x^3 y^2 + 32 x y^2\)[/tex]
1. Identify the GCF: The GCF of [tex]\(8 x^3 y^2\)[/tex] and [tex]\(32 x y^2\)[/tex] is [tex]\(8 x y^2\)[/tex].
2. Factor out the GCF: [tex]\(8 x y^2(x^2 + 4)\)[/tex]
So, the expression is [tex]\(8 x y^2(x^2 + 4)\)[/tex].
#### h. [tex]\(y(3 x - 2) - (3 x - 2)\)[/tex]
1. Identify the GCF: The GCF of [tex]\(y(3 x - 2)\)[/tex] and [tex]\(-(3 x - 2)\)[/tex] is [tex]\((3 x - 2)\)[/tex].
2. Factor out the GCF: [tex]\((3 x - 2)(y - 1)\)[/tex]
So, the expression is [tex]\((3 x - 2)(y - 1)\)[/tex].
#### i. [tex]\(5 x(x - 7) + 3(x - 7)\)[/tex]
1. Identify the GCF: The GCF of [tex]\(5 x(x - 7)\)[/tex] and [tex]\(3(x - 7)\)[/tex] is [tex]\((x - 7)\)[/tex].
2. Factor out the GCF: [tex]\((x - 7)(5 x + 3)\)[/tex]
So, the expression is [tex]\((x - 7)(5 x + 3)\)[/tex].
#### j. [tex]\(4 b(3 a b - 2) + 2(3 a b - 2) - 5 a(3 a b - 2)\)[/tex]
1. Identify the GCF: The GCF of all three terms is [tex]\((3 a b - 2)\)[/tex].
2. Factor out the GCF: [tex]\((3 a b - 2)(4 b + 2 - 5 a)\)[/tex]
So, the expression is [tex]\((3 a b - 2)(4 b + 2 - 5 a)\)[/tex].
### 2. Factor by Grouping
#### a. [tex]\(x y - 2 x + 7 y - 14\)[/tex]
1. Group terms: [tex]\( (x y - 2 x) + (7 y - 14) \)[/tex]
2. Factor out the GCF from each group: [tex]\( x(y - 2) + 7(y - 2) \)[/tex]
3. Factor out the common term [tex]\((y - 2)\)[/tex]: [tex]\( (x + 7)(y - 2) \)[/tex]
So, the expression is [tex]\((x + 7)(y - 2)\)[/tex].
#### b. [tex]\(x^3 - 8 x^2 + 3 x - 24\)[/tex]
1. Group terms: [tex]\( (x^3 - 8 x^2) + (3 x - 24) \)[/tex]
2. Factor out the GCF from each group: [tex]\( x^2(x - 8) + 3(x - 8) \)[/tex]
3. Factor out the common term [tex]\((x - 8)\)[/tex]: [tex]\( (x^2 + 3)(x - 8) \)[/tex]
So, the expression is [tex]\((x^2 + 3)(x - 8)\)[/tex].
#### c. [tex]\(a d + 3 a - d^2 - 3 d\)[/tex]
1. Group terms: [tex]\( (a d + 3 a) - (d^2 + 3 d) \)[/tex]
2. Factor out the GCF from each group: [tex]\( a(d + 3) - d(d + 3) \)[/tex]
3. Factor out the common term [tex]\((d + 3)\)[/tex]: [tex]\( (a - d)(d + 3) \)[/tex]
So, the expression is [tex]\((a - d)(d + 3)\)[/tex].
#### d. [tex]\(x y + 5 x - 5 y - 25\)[/tex]
1. Group terms: [tex]\( (x y + 5 x) - (5 y + 25) \)[/tex]
2. Factor out the GCF from each group: [tex]\( x(y + 5) - 5(y + 5) \)[/tex]
3. Factor out the common term [tex]\((y + 5)\)[/tex]: [tex]\( (x - 5)(y + 5) \)[/tex]
So, the expression is [tex]\((x - 5)(y + 5)\)[/tex].
#### e. [tex]\(35 - 5 a + 7 a - a^2\)[/tex]
1. Rearrange and group terms: [tex]\( (-a^2 + 7 a) + (35 - 5 a) \rightarrow (7 a - a^2) + (35 - 5 a) \)[/tex]
2. Factor out the GCF from each group: [tex]\( a(7 - a) + 5(7 - a) \)[/tex]
3. Factor out the common term [tex]\((7 - a)\)[/tex]: [tex]\( (a + 5)(7 - a) \)[/tex]
So, the expression is [tex]\( (a + 5)(7 - a) \)[/tex].
That's all. These are the factored forms of each given expression.
### 1. Remove the greatest common factor (GCF) and express each expression as the product of a GCF and a polynomial.
#### a. [tex]\(7 b^2 + 7 b\)[/tex]
1. Identify the GCF: The GCF of [tex]\(7 b^2\)[/tex] and [tex]\(7 b\)[/tex] is [tex]\(7 b\)[/tex].
2. Factor out the GCF: [tex]\(7 b(b + 1)\)[/tex]
So, the expression is [tex]\(7 b(b + 1)\)[/tex].
#### b. [tex]\(24 a b - 18 a b^2\)[/tex]
1. Identify the GCF: The GCF of [tex]\(24 a b\)[/tex] and [tex]\(18 a b^2\)[/tex] is [tex]\(6 a b\)[/tex].
2. Factor out the GCF: [tex]\(6 a b(4 - 3 b)\)[/tex]
So, the expression is [tex]\(6 a b(4 - 3 b)\)[/tex].
#### c. [tex]\(5 a^2 b^2 - 35 a b\)[/tex]
1. Identify the GCF: The GCF of [tex]\(5 a^2 b^2\)[/tex] and [tex]\(35 a b\)[/tex] is [tex]\(5 a b\)[/tex].
2. Factor out the GCF: [tex]\(5 a b(a b - 7)\)[/tex]
So, the expression is [tex]\(5 a b(a b - 7)\)[/tex].
#### d. [tex]\(8 x^3 - 10 x^2 - 14 x\)[/tex]
1. Identify the GCF: The GCF of [tex]\(8 x^3\)[/tex], [tex]\(10 x^2\)[/tex], and [tex]\(14 x\)[/tex] is [tex]\(2 x\)[/tex].
2. Factor out the GCF: [tex]\(2 x(4 x^2 - 5 x - 7)\)[/tex]
So, the expression is [tex]\(2 x(4 x^2 - 5 x - 7)\)[/tex].
#### e. [tex]\(36 x^6 + 45 x^4 - 18 x^2\)[/tex]
1. Identify the GCF: The GCF of [tex]\(36 x^6\)[/tex], [tex]\(45 x^4\)[/tex], and [tex]\(18 x^2\)[/tex] is [tex]\(9 x^2\)[/tex].
2. Factor out the GCF: [tex]\(9 x^2(4 x^4 + 5 x^2 - 2)\)[/tex]
So, the expression is [tex]\(9 x^2(4 x^4 + 5 x^2 - 2)\)[/tex].
#### f. [tex]\(40 a^2 - 16 a b - 24 a\)[/tex]
1. Identify the GCF: The GCF of [tex]\(40 a^2\)[/tex], [tex]\(16 a b\)[/tex], and [tex]\(24 a\)[/tex] is [tex]\(8 a\)[/tex].
2. Factor out the GCF: [tex]\(8 a(5 a - 2 b - 3)\)[/tex]
So, the expression is [tex]\(8 a(5 a - 2 b - 3)\)[/tex].
#### g. [tex]\(8 x^3 y^2 + 32 x y^2\)[/tex]
1. Identify the GCF: The GCF of [tex]\(8 x^3 y^2\)[/tex] and [tex]\(32 x y^2\)[/tex] is [tex]\(8 x y^2\)[/tex].
2. Factor out the GCF: [tex]\(8 x y^2(x^2 + 4)\)[/tex]
So, the expression is [tex]\(8 x y^2(x^2 + 4)\)[/tex].
#### h. [tex]\(y(3 x - 2) - (3 x - 2)\)[/tex]
1. Identify the GCF: The GCF of [tex]\(y(3 x - 2)\)[/tex] and [tex]\(-(3 x - 2)\)[/tex] is [tex]\((3 x - 2)\)[/tex].
2. Factor out the GCF: [tex]\((3 x - 2)(y - 1)\)[/tex]
So, the expression is [tex]\((3 x - 2)(y - 1)\)[/tex].
#### i. [tex]\(5 x(x - 7) + 3(x - 7)\)[/tex]
1. Identify the GCF: The GCF of [tex]\(5 x(x - 7)\)[/tex] and [tex]\(3(x - 7)\)[/tex] is [tex]\((x - 7)\)[/tex].
2. Factor out the GCF: [tex]\((x - 7)(5 x + 3)\)[/tex]
So, the expression is [tex]\((x - 7)(5 x + 3)\)[/tex].
#### j. [tex]\(4 b(3 a b - 2) + 2(3 a b - 2) - 5 a(3 a b - 2)\)[/tex]
1. Identify the GCF: The GCF of all three terms is [tex]\((3 a b - 2)\)[/tex].
2. Factor out the GCF: [tex]\((3 a b - 2)(4 b + 2 - 5 a)\)[/tex]
So, the expression is [tex]\((3 a b - 2)(4 b + 2 - 5 a)\)[/tex].
### 2. Factor by Grouping
#### a. [tex]\(x y - 2 x + 7 y - 14\)[/tex]
1. Group terms: [tex]\( (x y - 2 x) + (7 y - 14) \)[/tex]
2. Factor out the GCF from each group: [tex]\( x(y - 2) + 7(y - 2) \)[/tex]
3. Factor out the common term [tex]\((y - 2)\)[/tex]: [tex]\( (x + 7)(y - 2) \)[/tex]
So, the expression is [tex]\((x + 7)(y - 2)\)[/tex].
#### b. [tex]\(x^3 - 8 x^2 + 3 x - 24\)[/tex]
1. Group terms: [tex]\( (x^3 - 8 x^2) + (3 x - 24) \)[/tex]
2. Factor out the GCF from each group: [tex]\( x^2(x - 8) + 3(x - 8) \)[/tex]
3. Factor out the common term [tex]\((x - 8)\)[/tex]: [tex]\( (x^2 + 3)(x - 8) \)[/tex]
So, the expression is [tex]\((x^2 + 3)(x - 8)\)[/tex].
#### c. [tex]\(a d + 3 a - d^2 - 3 d\)[/tex]
1. Group terms: [tex]\( (a d + 3 a) - (d^2 + 3 d) \)[/tex]
2. Factor out the GCF from each group: [tex]\( a(d + 3) - d(d + 3) \)[/tex]
3. Factor out the common term [tex]\((d + 3)\)[/tex]: [tex]\( (a - d)(d + 3) \)[/tex]
So, the expression is [tex]\((a - d)(d + 3)\)[/tex].
#### d. [tex]\(x y + 5 x - 5 y - 25\)[/tex]
1. Group terms: [tex]\( (x y + 5 x) - (5 y + 25) \)[/tex]
2. Factor out the GCF from each group: [tex]\( x(y + 5) - 5(y + 5) \)[/tex]
3. Factor out the common term [tex]\((y + 5)\)[/tex]: [tex]\( (x - 5)(y + 5) \)[/tex]
So, the expression is [tex]\((x - 5)(y + 5)\)[/tex].
#### e. [tex]\(35 - 5 a + 7 a - a^2\)[/tex]
1. Rearrange and group terms: [tex]\( (-a^2 + 7 a) + (35 - 5 a) \rightarrow (7 a - a^2) + (35 - 5 a) \)[/tex]
2. Factor out the GCF from each group: [tex]\( a(7 - a) + 5(7 - a) \)[/tex]
3. Factor out the common term [tex]\((7 - a)\)[/tex]: [tex]\( (a + 5)(7 - a) \)[/tex]
So, the expression is [tex]\( (a + 5)(7 - a) \)[/tex].
That's all. These are the factored forms of each given expression.
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
